cs.GT

arXiv:2409.01963v3 Announce Type: replace Abstract: We study the problem of computing \emph{fair} divisions of a set of indivisible goods among agents with \emph{additive} valuations. For the past many decades, the literature has explored various notions of fairness, that can be primarily seen as either having \emph{envy-based} or \emph{share-based} lens. For the discrete setting of resource-allocation problems, \emph{envy-free up to any good} (EFX) and \emph{maximin share} (MMS) are widely considered as the flag-bearers of fairness notions in the above two categories, thereby capturing different aspects of fairness herein. Due to lack of existence results of these notions and the fact that a good approximation of EFX or MMS does not imply particularly strong guarantees of the other, it becomes important to understand the compatibility of EFX and MMS allocations with one another. In this work, we identify a novel way to simultaneously achieve MMS guarantees with EFX/EF1 notions of fairness, while beating the best known approximation factors [Chaudhury et al., 2021, Amanatidis et al., 2020]. Our main contribution is to constructively prove the existence of (i) a partial allocation that is both $2/3$-MMS and EFX, and (ii) a complete allocation that is both $2/3$-MMS and EF1. Our algorithms run in pseudo-polynomial time if the approximation factor for MMS is relaxed to $2/3-\varepsilon$ for any constant $\varepsilon > 0$ and in polynomial time if, in addition, the EFX (or EF1) guarantee is relaxed to $(1-\delta)$-EFX (or $(1-\delta)$-EF1) for any constant $\delta>0$. In particular, we improve from the best approximation factor known prior to our work, which computes partial allocations that are $1/2$-MMS and EFX in pseudo-polynomial time [Chaudhury et al., 2021].

arXiv:2410.17517v2 Announce Type: replace Abstract: Swarm intelligence (SI) explores how large groups of simple individuals (e.g., insects, fish, birds) collaborate to produce complex behaviors, exemplifying that the whole is greater than the sum of its parts. A fundamental task in SI is Collective Decision-Making (CDM), where a group selects the best option among several alternatives, such as choosing an optimal foraging site. In this work, we demonstrate a theoretical and empirical equivalence between CDM and single-agent reinforcement learning (RL) in multi-armed bandit problems, utilizing concepts from opinion dynamics, evolutionary game theory, and RL. This equivalence bridges the gap between SI and RL and leads us to introduce a novel abstract RL update rule called Maynard-Cross Learning. Additionally, it provides a new population-based perspective on common RL practices like learning rate adjustment and batching. Our findings enable cross-disciplinary fertilization between RL and SI, allowing techniques from one field to enhance the understanding and methodologies of the other.

arXiv:2210.02773v3 Announce Type: replace Abstract: In a two-player zero-sum graph game, the players move a token throughout a graph to produce an infinite play, which determines the winner of the game. Bidding games are graph games in which in each turn, an auction (bidding) determines which player moves the token: the players have budgets, and in each turn, both players simultaneously submit bids that do not exceed their available budgets, the higher bidder moves the token, and pays the bid to the lower bidder (called Richman bidding). We focus on discrete-bidding games, in which, motivated by practical applications, the granularity of the players' bids is restricted, e.g., bids must be given in cents. A central quantity in bidding games is threshold budgets: a necessary and sufficient initial budget for winning the game. Previously, thresholds were shown to exist in parity games, but their structure was only understood for reachability games. Moreover, the previously-known algorithms have a worst-case exponential running time for both reachability and parity objectives, and output strategies that use exponential memory. We describe two algorithms for finding threshold budgets in parity discrete-bidding games. The first is a fixed-point algorithm. It reveals, for the first time, the structure of threshold budgets in parity discrete-bidding games. Based on this structure, we develop a second algorithm that shows that the problem of finding threshold budgets is in NP and coNP for both reachability and parity objectives. Moreover, our algorithm constructs strategies that use only linear memory.

arXiv:2501.04180v2 Announce Type: replace Abstract: Games have been vital test beds for the rapid development of Agent-based research. Remarkable progress has been achieved in the past, but it is unclear if the findings equip for real-world problems. While pressure grows, some of the most critical ecological challenges can find mitigation and prevention solutions through technology and its applications. Most real-world domains include multi-agent scenarios and require machine-machine and human-machine collaboration. Open-source environments have not advanced and are often toy scenarios, too abstract or not suitable for multi-agent research. By mimicking real-world problems and increasing the complexity of environments, we hope to advance state-of-the-art multi-agent research and inspire researchers to work on immediate real-world problems. Here, we present HIVEX, an environment suite to benchmark multi-agent research focusing on ecological challenges. HIVEX includes the following environments: Wind Farm Control, Wildfire Resource Management, Drone-Based Reforestation, Ocean Plastic Collection, and Aerial Wildfire Suppression. We provide environments, training examples, and baselines for the main and sub-tasks. All trained models resulting from the experiments of this work are hosted on Hugging Face. We also provide a leaderboard on Hugging Face and encourage the community to submit models trained on our environment suite.

arXiv:2312.03121v3 Announce Type: replace Abstract: We argue that many general evaluation problems can be viewed through the lens of voting theory. Each task is interpreted as a separate voter, which requires only ordinal rankings or pairwise comparisons of agents to produce an overall evaluation. By viewing the aggregator as a social welfare function, we are able to leverage centuries of research in social choice theory to derive principled evaluation frameworks with axiomatic foundations. These evaluations are interpretable and flexible, while avoiding many of the problems currently facing cross-task evaluation. We apply this Voting-as-Evaluation (VasE) framework across multiple settings, including reinforcement learning, large language models, and humans. In practice, we observe that VasE can be more robust than popular evaluation frameworks (Elo and Nash averaging), discovers properties in the evaluation data not evident from scores alone, and can predict outcomes better than Elo in a complex seven-player game. We identify one particular approach, maximal lotteries, that satisfies important consistency properties relevant to evaluation, is computationally efficient (polynomial in the size of the evaluation data), and identifies game-theoretic cycles.

arXiv:2501.10884v1 Announce Type: new Abstract: We propose a new algorithm that finds an $\varepsilon$-approximate fixed point of a smooth function from the $n$-dimensional $\ell_2$ unit ball to itself. We use the general framework of finding approximate solutions to a variational inequality, a problem that subsumes fixed point computation and the computation of a Nash Equilibrium. The algorithm's runtime is bounded by $e^{O(n)}/\varepsilon$, under the smoothed-analysis framework. This is the first known algorithm in such a generality whose runtime is faster than $(1/\varepsilon)^{O(n)}$, which is a time that suffices for an exhaustive search. We complement this result with a lower bound of $e^{\Omega(n)}$ on the query complexity for finding an $O(1)$-approximate fixed point on the unit ball, which holds even in the smoothed-analysis model, yet without the assumption that the function is smooth. Existing lower bounds are only known for the hypercube, and adapting them to the ball does not give non-trivial results even for finding $O(1/\sqrt{n})$-approximate fixed points.

arXiv:2501.11024v1 Announce Type: new Abstract: Networks significantly influence social, economic, and organizational outcomes, with centrality measures serving as crucial tools to capture the importance of individual nodes. This paper introduces Laplacian Eigenvector Centrality (LEC), a novel framework for network analysis based on spectral graph theory and the eigendecomposition of the Laplacian matrix. A distinctive feature of LEC is its adjustable parameter, the LEC order, which enables researchers to control and assess the scope of centrality measurement using the Laplacian spectrum. Using random graph models, LEC demonstrates robustness and scalability across diverse network structures. We connect LEC to equilibrium responses to external shocks in an economic model, showing how LEC quantifies agents' roles in attenuating shocks and facilitating coordinated responses through quadratic optimization. Finally, we apply LEC to the study of microfinance diffusion, illustrating how it complements classical centrality measures, such as eigenvector and Katz-Bonacich centralities, by capturing distinctive aspects of node positions within the network.

arXiv:2501.12199v1 Announce Type: new Abstract: Despite its groundbreaking success, multi-agent reinforcement learning (MARL) still suffers from instability and nonstationarity. Replicator dynamics, the most well-known model from evolutionary game theory (EGT), provide a theoretical framework for the convergence of the trajectories to Nash equilibria and, as a result, have been used to ensure formal guarantees for MARL algorithms in stable game settings. However, they exhibit the opposite behavior in other settings, which poses the problem of finding alternatives to ensure convergence. In contrast, innovative dynamics, such as the Brown-von Neumann-Nash (BNN) or Smith, result in periodic trajectories with the potential to approximate Nash equilibria. Yet, no MARL algorithms based on these dynamics have been proposed. In response to this challenge, we develop a novel experience replay-based MARL algorithm that incorporates revision protocols as tunable hyperparameters. We demonstrate, by appropriately adjusting the revision protocols, that the behavior of our algorithm mirrors the trajectories resulting from these dynamics. Importantly, our contribution provides a framework capable of extending the theoretical guarantees of MARL algorithms beyond replicator dynamics. Finally, we corroborate our theoretical findings with empirical results.

arXiv:2406.10631v2 Announce Type: replace Abstract: Self-play via online learning is one of the premier ways to solve large-scale two-player zero-sum games, both in theory and practice. Particularly popular algorithms include optimistic multiplicative weights update (OMWU) and optimistic gradient-descent-ascent (OGDA). While both algorithms enjoy $O(1/T)$ ergodic convergence to Nash equilibrium in two-player zero-sum games, OMWU offers several advantages including logarithmic dependence on the size of the payoff matrix and $\widetilde{O}(1/T)$ convergence to coarse correlated equilibria even in general-sum games. However, in terms of last-iterate convergence in two-player zero-sum games, an increasingly popular topic in this area, OGDA guarantees that the duality gap shrinks at a rate of $O(1/\sqrt{T})$, while the best existing last-iterate convergence for OMWU depends on some game-dependent constant that could be arbitrarily large. This begs the question: is this potentially slow last-iterate convergence an inherent disadvantage of OMWU, or is the current analysis too loose? Somewhat surprisingly, we show that the former is true. More generally, we prove that a broad class of algorithms that do not forget the past quickly all suffer the same issue: for any arbitrarily small $\delta>0$, there exists a $2\times 2$ matrix game such that the algorithm admits a constant duality gap even after $1/\delta$ rounds. This class of algorithms includes OMWU and other standard optimistic follow-the-regularized-leader algorithms.

arXiv:2412.14570v2 Announce Type: replace Abstract: In Tennenholtz's program equilibrium, players of a game submit programs to play on their behalf. Each program receives the other programs' source code and outputs an action. This can model interactions involving AI agents, mutually transparent institutions, or commitments. Tennenholtz (2004) proves a folk theorem for program games, but the equilibria constructed are very brittle. We therefore consider simulation-based programs -- i.e., programs that work by running opponents' programs. These are relatively robust (in particular, two programs that act the same are treated the same) and are more practical than proof-based approaches. Oesterheld's (2019) $\epsilon$Grounded$\pi$Bot is such an approach. Unfortunately, it is not generally applicable to games of three or more players, and only allows for a limited range of equilibria in two player games. In this paper, we propose a generalisation to Oesterheld's (2019) $\epsilon$Grounded$\pi$Bot. We prove a folk theorem for our programs in a setting with access to a shared source of randomness. We then characterise their equilibria in a setting without shared randomness. Both with and without shared randomness, we achieve a much wider range of equilibria than Oesterheld's (2019) $\epsilon$Grounded$\pi$Bot. Finally, we explore the limits of simulation-based program equilibrium, showing that the Tennenholtz folk theorem cannot be attained by simulation-based programs without access to shared randomness.

arXiv:2501.10388v1 Announce Type: new Abstract: The emergence of Large Language Models has fundamentally transformed the capabilities of AI agents, enabling a new class of autonomous agents capable of interacting with their environment through dynamic code generation and execution. These agents possess the theoretical capacity to operate as independent economic actors within digital markets, offering unprecedented potential for value creation through their distinct advantages in operational continuity, perfect replication, and distributed learning capabilities. However, contemporary digital infrastructure, architected primarily for human interaction, presents significant barriers to their participation. This work presents a systematic analysis of the infrastructure requirements necessary for AI agents to function as autonomous participants in digital markets. We examine four key areas - identity and authorization, service discovery, interfaces, and payment systems - to show how existing infrastructure actively impedes agent participation. We argue that addressing these infrastructure challenges represents more than a technical imperative; it constitutes a fundamental step toward enabling new forms of economic organization. Much as traditional markets enable human intelligence to coordinate complex activities beyond individual capability, markets incorporating AI agents could dramatically enhance economic efficiency through continuous operation, perfect information sharing, and rapid adaptation to changing conditions. The infrastructure challenges identified in this work represent key barriers to realizing this potential.

arXiv:2501.10464v1 Announce Type: new Abstract: We study the problem of adapting to a known sub-rational opponent during online play while remaining robust to rational opponents. We focus on large imperfect-information (zero-sum) games, which makes it impossible to inspect the whole game tree at once and necessitates the use of depth-limited search. However, all existing methods assume rational play beyond the depth-limit, which only allows them to adapt a very limited portion of the opponent's behaviour. We propose an algorithm Adapting Beyond Depth-limit (ABD) that uses a strategy-portfolio approach - which we refer to as matrix-valued states - for depth-limited search. This allows the algorithm to fully utilise all information about the opponent model, making it the first robust-adaptation method to be able to do so in large imperfect-information games. As an additional benefit, the use of matrix-valued states makes the algorithm simpler than traditional methods based on optimal value functions. Our experimental results in poker and battleship show that ABD yields more than a twofold increase in utility when facing opponents who make mistakes beyond the depth limit and also delivers significant improvements in utility and safety against randomly generated opponents.

arXiv:2501.12015v1 Announce Type: new Abstract: In multiwinner approval voting, forming a committee that proportionally represents voters' approval ballots is an essential task. The notion of justified representation (JR) demands that any large "cohesive" group of voters should be proportionally "represented". The "cohesiveness" is defined in different ways; two common ways are the following: (C1) demands that the group unanimously approves a set of candidates proportional to its size, while (C2) requires each member to approve at least a fixed fraction of such a set. Similarly, "representation" have been considered in different ways: (R1) the coalition's collective utility from the winning set exceeds that of any proportionally sized alternative, and (R2) for any proportionally sized alternative, at least one member of the coalition derives less utility from it than from the winning set. Three of the four possible combinations have been extensively studied: (C1)-(R1) defines Proportional Justified Representation (PJR), (C1)-(R2) defines Extended Justified Representation (EJR), (C2)-(R2) defines Full Justified Representation (FJR). All three have merits, but also drawbacks. PJR is the weakest notion, and perhaps not sufficiently demanding; EJR may not be compatible with perfect representation; and it is open whether a committee satisfying FJR can be found efficiently. We study the combination (C2)-(R1), which we call Full Proportional Justified Representation (FPJR). We investigate FPJR's properties and find that it shares PJR's advantages over EJR: several proportionality axioms (e.g. priceability, perfect representation) imply FPJR and PJR but not EJR. We also find that efficient rules like the greedy Monroe rule and the method of equal shares satisfy FPJR, matching a key advantage of EJR over FJR. However, the Proportional Approval Voting (PAV) rule may violate FPJR, so neither of EJR and FPJR implies the other.

arXiv:2501.11897v1 Announce Type: new Abstract: We formulate and study a general time-varying multi-agent system where players repeatedly compete under incomplete information. Our work is motivated by scenarios commonly observed in online advertising and retail marketplaces, where agents and platform designers optimize algorithmic decision-making in dynamic competitive settings. In these systems, no-regret algorithms that provide guarantees relative to \emph{static} benchmarks can perform poorly and the distributions of play that emerge from their interaction do not correspond anymore to static solution concepts such as coarse correlated equilibria. Instead, we analyze the interaction of \textit{dynamic benchmark} consistent policies that have performance guarantees relative to \emph{dynamic} sequences of actions, and through a novel \textit{tracking error} notion we delineate when their empirical joint distribution of play can approximate an evolving sequence of static equilibria. In systems that change sufficiently slowly (sub-linearly in the horizon length), we show that the resulting distributions of play approximate the sequence of coarse correlated equilibria, and apply this result to establish improved welfare bounds for smooth games. On a similar vein, we formulate internal dynamic benchmark consistent policies and establish that they approximate sequences of correlated equilibria. Our findings therefore suggest that, in a broad range of multi-agent systems where non-stationarity is prevalent, algorithms designed to compete with dynamic benchmarks can improve both individual and welfare guarantees, and their emerging dynamics approximate a sequence of static equilibrium outcomes.

arXiv:2403.08051v2 Announce Type: replace Abstract: Rent division is the well-studied problem of fairly assigning rooms and dividing rent among a set of roommates within a single apartment. A shortcoming of existing solutions is that renters are assumed to be considering apartments in isolation, whereas in reality, renters can choose among multiple apartments. In this paper, we generalize the rent division problem to the multi-apartment setting, where the goal is to both fairly choose an apartment among a set of alternatives and fairly assign rooms and rents within the chosen apartment. Our main contribution is a generalization of envy-freeness called negotiated envy-freeness. We show that a solution satisfying negotiated envy-freeness is guaranteed to exist and that it is possible to optimize over all negotiated envy-free solutions in polynomial time. We also define an even stronger fairness notion called universal envy-freeness and study its existence when values are drawn randomly.

arXiv:2401.04318v2 Announce Type: replace Abstract: We study the problem of allocating indivisible items on a path among agents. The objective is to find a fair and efficient allocation in which each agent's bundle forms a contiguous block on the line. We say that an instance is \emph{$(a, b)$-sparse} if each agent values at most $a$ items positively and each item is valued positively by at most $b$ agents. We demonstrate that, even when the valuations are binary additive, deciding whether every item can be allocated to an agent who wants it is NP-complete for the $(4,3)$-sparse instances. Consequently, we provide two fixed-parameter tractable (FPT) algorithms for maximizing utilitarian social welfare, with respect to the number of agents and the number of items. Additionally, we present a $2$-approximation algorithm for the special case when the valuations are binary additive, and the maximum utility is equal to the number of items. Also, we provide a $1/a$-approximation algorithm for the $(a,b)$-sparse instances. Furthermore, we establish that deciding whether the maximum egalitarian social welfare is at least $2$ or at most $1$ is NP-complete for the $(6,3)$-sparse instances, even when the valuations are binary additive. We present a $1/a$-approximation algorithm for maximizing egalitarian social welfare for the $(a,b)$-sparse instances. Besides, we give two FPT algorithms for maximizing egalitarian social welfare in terms of the number of agents and the number of items. We also explore the case where the order of the blocks of items allocated to the agents is predetermined. In this case, we show that both maximum utilitarian social welfare and egalitarian social welfare can be computed in polynomial time. However, we determine that checking the existence of an EF1 allocation is NP-complete, even when the valuations are binary additive.

arXiv:2403.15307v3 Announce Type: replace Abstract: Today we rely on networks that are created and maintained by smart devices. For such networks, there is no governing central authority but instead the network structure is shaped by the decisions of selfish intelligent agents. A key property of such communication networks is that they should be easy to navigate for routing data. For this, a common approach is greedy routing, where every device simply routes data to a neighbor that is closer to the respective destination. Networks of intelligent agents can be analyzed via a game-theoretic approach and in the last decades many variants of network creation games have been proposed and analyzed. In this paper we present the first game-theoretic network creation model that incorporates greedy routing, i.e., the strategic agents in our model are embedded in some metric space and strive for creating a network among themselves where all-pairs greedy routing is enabled. Besides this, the agents optimize their connection quality within the created network by aiming for greedy routing paths with low stretch. For our model, we analyze the existence of (approximate)-equilibria and the computational hardness in different underlying metric spaces. E.g., we characterize the set of equilibria in 1-2-metrics and tree metrics and show that Nash equilibria always exist. For Euclidean space, the setting which is most relevant in practice, we prove that equilibria are not guaranteed to exist but that the well-known $\Theta$-graph construction yields networks having a low stretch that are game-theoretically almost stable. For general metric spaces, we show that approximate equilibria exist where the approximation factor depends on the cost of maintaining any link.

arXiv:2409.04669v4 Announce Type: replace Abstract: Matching algorithms have demonstrated great success in several practical applications, but they often require centralized coordination and plentiful information. In many modern online marketplaces, agents must independently seek out and match with another using little to no information. For these kinds of settings, can we design decentralized, limited-information matching algorithms that preserve the desirable properties of standard centralized techniques? In this work, we constructively answer this question in the affirmative. We model a two-sided matching market as a game consisting of two disjoint sets of agents, referred to as proposers and acceptors, each of whom seeks to match with their most preferable partner on the opposite side of the market. However, each proposer has no knowledge of their own preferences, so they must learn their preferences while forming matches in the market. We present a simple online learning rule that guarantees a strong notion of probabilistic convergence to the welfare-maximizing equilibrium of the game, referred to as the proposer-optimal stable match. To the best of our knowledge, this represents the first completely decoupled, communication-free algorithm that guarantees probabilistic convergence to an optimal stable match, irrespective of the structure of the matching market.

arXiv:2501.06506v1 Announce Type: new Abstract: A Latin square is an $n \times n$ matrix filled with $n$ distinct symbols, each of which appears exactly once in each row and exactly once in each column. We introduce a problem of allocating $n$ indivisible items among $n$ agents over $n$ rounds while satisfying the Latin square constraint. This constraint ensures that each agent receives no more than one item per round and receives each item at most once. Each agent has an additive valuation on the item--round pairs. Real-world applications like scheduling, resource management, and experimental design require the Latin square constraint to satisfy fairness or balancedness in allocation. Our goal is to find a partial or complete allocation that maximizes the sum of the agents' valuations (utilitarian social welfare) or the minimum of the agents' valuations (egalitarian social welfare). For the problem of maximizing utilitarian social welfare, we prove NP-hardness even when the valuations are binary additive. We then provide $(1-1/e)$ and $(1-1/e)/4$-approximation algorithms for partial and complete settings, respectively. Additionally, we present fixed-parameter tractable (FPT) algorithms with respect to the order of Latin square and the optimum value for both partial and complete settings. For the problem of maximizing egalitarian social welfare, we establish that deciding whether the optimum value is at most $1$ or at least $2$ is NP-hard for both the partial and complete settings, even when the valuations are binary. Furthermore, we demonstrate that checking the existence of a complete allocation that satisfies each of envy-free, proportional, equitable, envy-free up to any good, proportional up to any good, or equitable up to any good is NP-hard, even when the valuations are identical.

arXiv:2501.06261v1 Announce Type: new Abstract: Class Activation Mapping (CAM) methods are widely used to visualize neural network decisions, yet their underlying mechanisms remain incompletely understood. To enhance the understanding of CAM methods and improve their explainability, we introduce the Content Reserved Game-theoretic (CRG) Explainer. This theoretical framework clarifies the theoretical foundations of GradCAM and HiResCAM by modeling the neural network prediction process as a cooperative game. Within this framework, we develop ShapleyCAM, a new method that leverages gradients and the Hessian matrix to provide more precise and theoretically grounded visual explanations. Due to the computational infeasibility of exact Shapley value calculation, ShapleyCAM employs a second-order Taylor expansion of the cooperative game's utility function to derive a closed-form expression. Additionally, we propose the Residual Softmax Target-Class (ReST) utility function to address the limitations of pre-softmax and post-softmax scores. Extensive experiments across 12 popular networks on the ImageNet validation set demonstrate the effectiveness of ShapleyCAM and its variants. Our findings not only advance CAM explainability but also bridge the gap between heuristic-driven CAM methods and compute-intensive Shapley value-based methods. The code is available at \url{https://github.com/caihuaiguang/pytorch-shapley-cam}.